On the straightening of every functor

We show that any functor between $\mathrm{\infty }$-categories can be straightened. More precisely, we show that for any $\mathrm{\infty }$-category $\mathcal{C}$, there is an equivalence between the $\mathrm{\infty }$-category $({\mathrm{C}\mathrm{a}\mathrm{t}}_{\mathrm{\infty }}{)}_{\mathrm{/}\mathcal{C}}$ of $\mathrm{\infty }$-categories over $\mathcal{C}$ and the $\mathrm{\infty }$-category of unital lax functors from $\mathcal{C}$ to the double $\mathrm{\infty }$-category $\mathrm{C}\mathrm{o}\mathrm{r}\mathrm{r}$ of correspondences. The proof relies on a certain universal property of the Morita category which is of independent interest.